#6347493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347491
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(18,24)(22,29)(23,30)(28,31);(2,9)(3,13)(8,10)(12,14)

Polycyclic presentation:

2^9|d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254689);